Question

How do you use the Squeeze Theorem to find `lim xcos(1/x)` as x approaches zero?

Answer 1

Use the fact that the cosine function is always between `-1` and `1`, implying that the given function is always between `-|x|` and `|x|`, which both go to zero as `x` goes to zero.

Explanation:

Let `f(x)=x cos(1/x)`, `g(x)=-|x|`, and `h(x)=|x|`. Since `-1 leq cos(1/x) leq 1` for all `x !=0`, it follows that `g(x) leq f(x) leq h(x)` for all `x !=0`.

But `lim_{x->0}g(x)=lim_{x->0}h(x)=0`. Therefore, the Squeeze Theorem can be use to conclude that `lim_{x->0}f(x)=lim_{x->0}x cos(1/x)=0`.